Strong measure zero sets, filters, and pointwise convergence. Marion Scheepers * The notion of a strong measure zero space (but not this terminology) was introduced in [1] by E. Borel. A separable metric space X has strong measure zero if there is for each sequence (ǫ n : n ∈ N) of positive real numbers a par- tition X = ∪ n X n such that for each n the diameter of X n is less than ǫ n . In [20] we found characterizations of the notion of a strong measure zero set in terms of certain selection principles applied to open covers, and also in terms of Ramseyan partition relations for certain families of open covers. Also, the property of having strong measure zero in all finite powers was characterized like this. Due to successes in using the concept of a filter on the set of natu- ral numbers to describe certain covering properties of sets of real numbers, as in [10], and of using the closure properties of function spaces, as in [17], the question arose whether these methods would also yield analogous descriptions of strong measure zero spaces. In the paper [9] we made a beginning in this di- rection by showing that for certain function spaces derived from a space X , the strong measure zero-ness of all finite powers of X is equivalent to certain closure properties of the associated function space. The function space associated with X was obtained by first associating with X a subspace T (X ) of the Alexandroff double of the closed unit interval [0,1], and by then taking the function space C s (T (X )) – this space differs from the usual space topologized by the topology of pointwise convergence in that we consider a coarser topology described in terms of a dense discrete subspace consisting of the isolated points of T (X ). We shall see below that the intermediate construction T (X ) can be avoided, and that the traditional topology of pointwise convergence is sufficient for the task of detecting the strong measure zero ness of the domain space. We shall also see that the filter description can be carried through here. To set the stage for the work to be done, we first introduce some necessary notation and concepts, and cite one of the principal results from [20] which will be used here. An open cover U of a space is an ω–cover if the space itself is not a member of U , but for each finite subset F of the space there is a U ∈U with F ⊆ U . Thus, let Y be a compact metric space, and let X be a subspace of Y . O: The collection of all open covers of Y ; D: The collection of all covers of X by sets open in Y ; * Supported by NSF grant DMS 95-05375 1